In a certain town 25% families own a phone and 15% own a car, 65% families own neither a phone nor a car. 2000 families own both a car and a phone.
Consider the following statements in this regard:
1. 10% families own both a car and a phone
2. 35% families own either a car or a phone
3. 40,000 families live in the town
Which of the following statements are correct?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 2 and 3
Let there be x families in the town. Let P and C denote the set of families using phone and car respectively.
Then,
$n(P) =\frac{25x}{100}=\frac{x}{4}, n(C)=\frac{15x}{100}=\frac{3x}{20}$
and, $n(\overline P∩\overline C)=\frac{65x}{100}=\frac{13x}{20}$
Also, $n(P∩C) = 2000$.
Now,
$n(\overline P∩\overline C)=\frac{13x}{20}$
$⇒n(\overline{P∪C})=\frac{13x}{20}$
$⇒n(U)-n(P∩C)=\frac{13x}{20}$
$⇒x-\{n(P)+n(C)-n(P∩C)\}=\frac{13x}{20}$
$⇒x-\left(\frac{x}{4}+\frac{3x}{20}-2000\right)=\frac{13x}{20}⇒\frac{x}{20}=2000⇒x=40000$
Thus, statement 3 is true.
10% of the total families in the town is 4000 and it is given that 2000 families own both a car and a phone. So, statement 1 is not true.
Now,
$n(P∪C)=n (P) + n (C) -n (P∩C)$
$⇒n(P∪C)=\frac{x}{4}+\frac{3x}{20}- 2000$
$⇒n(P∪C)=10000+ 6000 - 2000 = 14000$
$⇒n(P∪C)$ = 35% of 40000
Thus, statement 2 is correct.